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Theorem brtrclfv 13677
Description: Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
brtrclfv (𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))
Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   𝑅,𝑟
Allowed substitution hint:   𝑉(𝑟)

Proof of Theorem brtrclfv
StepHypRef Expression
1 trclfv 13675 . . 3 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
21breqd 4624 . 2 (𝑅𝑉 → (𝐴(t+‘𝑅)𝐵𝐴 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝐵))
3 brintclab 13676 . . 3 (𝐴 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
4 df-br 4614 . . . . 5 (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
54imbi2i 326 . . . 4 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
65albii 1744 . . 3 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
73, 6bitr4i 267 . 2 (𝐴 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))
82, 7syl6bb 276 1 (𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1987  {cab 2607  wss 3555  cop 4154   cint 4440   class class class wbr 4613  ccom 5078  cfv 5847  t+ctcl 13658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-trcl 13660
This theorem is referenced by:  brcnvtrclfv  13678  brtrclfvcnv  13679  trclfvcotr  13684
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